3.3, 3.6 prove theorems about perpendicular lines

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3.3, 3.6 Prove Theorems about Perpendicular Lines

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3.3, 3.6 Prove Theorems about Perpendicular Lines

Proving Conditional Statements (if-then)Name Math Statement True/False

Conditional P → Q If a set of angles are a linear pair, then they are supplementary

True

Converse Q → P If a set of angles are supplementary, then they are a linear pair.

False

Inverse ~P →~Q If a set of angles are not a linear pair, then they are not supplementary

False

Contrapositive ~Q →~P If a set of angles are not supplementary, then they are not a linear pair.

True

Biconditional P↔Q A set of angles are a linear pair IFF they are supplementary

FALSE since Conditional and Converse not both true

If I know corresponding and alternate interior/exterior angles are congruent , I can prove lines are parallel.

Theorems/Postulates CONVERSE

• Converse of Corresponding Angles Postulate:– If 2 lines are cut by a transversal, and a pair of

corresponding angles are congruent, then the 2 lines are parallel

• THEOREMS: If two lines are cut by a transversal and a pair of– alternate interior angles are congruent, then the

lines are parallel– alternate exterior angles are congruent, then the

lines are parallel– consecutive interior angles are supplementary,

then the lines are parallel

Transitive Property of Lines

Parallel Lines Properties

Piecing it Together

Two Column Proof

Paragraph Proof

Conclusion:Two lines are cut by a transversal. How can you prove the

lines are parallel?

Show that either a pair of alternate interior angles, a pair of corresponding angles, or a pair of alternate exterior angles is congruent, or show that a pair of consecutive interior angles is supplementary.

Write 3 bi-conditional Statements using the theorems we discussed today

Coordinate Proof• 1) Prove that quadrilateral A(1,2), B(2,5),

C(5,7) and D(4,4) is a parallelogram by using slopes.

• Prove that A(1,1), B(4,4), C(6,2) are the vertices of a right triangle.

• 5) Prove that A(-3,2), B(-2,6), C(2,7)and D(1,3) is a rhombus.

• Prove that A(4,-1), B(5,6), C(1,3) is an isosceles right triangle.

Perpendicular Problems

Perpendicular Discussion

• Perpendicular and Linear Pairs

• Perpendicular Lines and Angles Formed

• Adjacent Acute Angles and Perpendicular Lines

• Transversal perpendicular to 1 line in a set of parallel lines

• Lines perpendicular to the same line

Congruent Angles Linear Pairs

Congruent Angles Linear Pairs

Perpendicular Lines Right Angles

Adjacent Acute Angles

Perpendicular Transversal Theorem

Lines Parallel to a Transversal Theorem